Dynamics of neighborhoods of points under a continuous mapping of an interval

Let $\{ I, f Z^{+} \}$ be a dynamical system induced by the continuous map $f$ of a closed bounded interval $I$ into itself.
In order to describe the dynamics of neighborhoods of points unstable under $f$, we suggest a notion of $\varepsilon \omega - {\rm set} \omega_{f, \varepsilon}(x)$ of a point $x$ as
the $\omega$-limit set of $\varepsilon$-neighborhood of $x$.
We investigate the association between the $\varepsilon \omega - {\rm set}$ and the domain of influence of a point. We also show that the domain of influence of an unstable point is always a cycle of intervals.
The results obtained can be directly applied in the theory of continuous time difference equations and similar equations.